By Klaus Jansen (auth.), Roberto Solis-Oba, Giuseppe Persiano (eds.)

This booklet constitutes the completely refereed post-proceedings of the ninth foreign Workshop on Approximation and on-line Algorithms, WAOA 2011, held in Saarbrücken, Germany, in September 2011. The 21 papers offered have been rigorously reviewed and chosen from forty eight submissions. the amount additionally includes a longer summary of the invited speak of Prof. Klaus Jansen. The Workshop on Approximation and on-line Algorithms makes a speciality of the layout and research of algorithms for on-line and computationally not easy difficulties. either sorts of difficulties have quite a few functions in a large choice of fields. subject matters of curiosity for WAOA 2011 have been: algorithmic online game concept, approximation periods, coloring and partitioning, aggressive research, computational finance, cuts and connectivity, geometric difficulties, inapproximability effects, mechanism layout, community layout, packing and masking, paradigms for layout and research of approximation and on-line algorithms, parameterized complexity, randomization suggestions and scheduling problems.

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**Extra info for Approximation and Online Algorithms: 9th International Workshop, WAOA 2011, Saarbrücken, Germany, September 8-9, 2011, Revised Selected Papers**

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Lemma 4. For a suﬃciently long input σ, if σ is d-slack (d > 0) and x1 , x2 , . . (xi ≤ a) satisfy inequality (1), the following inequality holds: ∞ xi 1 + i=1 . d ad And the equality holds for d ≥ h − 1 where h = min{i | xi = 0}. RDRA (σ) ≤ 1 + (2) Proof. To analyze the worst case input, we deﬁne σ(k) as an input in which one busy request arrives after (k−1) slack requests, where k = 1, 2, . . , is any positive integer. Then we ﬁnd that all the worst case input instances are described as the combination of σ(k) by Lemma 2.

Marb´ an, C. Rutten, and T. Vredeveld A formal proof of this theorem is given in the full version of the paper. In order to give this proof, we need a lower bound on the performance of -SEPT. To obtain this bound, we adjust the worst case instance of SEPT, given in Lemma 3. In that instance, we set our parameters in such a way that E [X1 ] is slightly less than E [X2 ]. Hence, SEPT starts processing all jobs of class J1 , followed by the jobs of class J2 . OPT however, starts processing a job from class J2 , since the distribution of Θ2 is ﬂat, making it is beneﬁcial to process a few jobs of the second class to get a better idea about the value of ϑ2 .

Tn be non-negative real values satisfying 0 ≤ t1 < · · · < tn representing the time of service of the device for n requests. An input is given as σ = t1 , t2 , . . , tn . We do nothing if the state is ON at ti (i = 1, . . , n), and should turn a device ON if it is OFF at that time. For a given input σ = t1 , t2 , . . , tn (n may be ∞), the action of an algorithm is determined by a sequence w1 , w2 , . . , wn , where wi is standby time after ith request is leaving. In other words, the problem is how to determine wi from t1 , t2 , .